Such a socketed lamp is known from U.S. Pat. No. 6,578,991B2. In the known lamp a construction for cooling a lamp vessel is disclosed. In general, cooling is required for lamps that are operated at relatively very high temperatures, for example special metal halide lamps, such as MSR lamps that are used for stage, studio, and theatre lighting. In particular, cooling is applied to lamp parts that are sensitive to oxidation, such as electrical current conductors at the location where they issue from the seals of the lamp vessel to the exterior. Therefore, in the known lamp a special construction is disclosed that aims at cooling only the seals. The construction comprises a large air chamber and a spout. It is a drawback of the known lamp that the construction to cool the seal is relatively bulky and expensive. Another drawback of the known lamp is that the cooling construction comprises many parts that have to be mounted and positioned onto the lamp vessel.